Instructor's guide

Orthogonal Series and Boundary Value Problems

Evans M. Harrell II*

*(c) Copyright 1994,1997 by Evans M. Harrell, II and James V. Herod. All rights reserved.

Either this chapter or the one on the heat equation may be done first.

Whichever chapter on PDEs is done first marks a very important point in the class, where the emphasis changes from orthogonal series to differential equations. Take the time required and do as many separations of variables as it takes for the students to get the idea. One idea: Do Dirichlet and Neumann in class and have them investigate periodic boundary conditions for exercises.

There is a link to a derivation of the wave equation for the vibrating string. This is based on Lagrangian mechanics, and I always preface it by reassuring students that they will not be responsible for Langangian mechanics. On the other hand, I have had very positive responses to it from my upper division and graduate engineering students.

The derivation is not the usual one, and has the advantage of bringing out exactly where the microscopic physics of the tension enters - there is a general potential energy contribution depending on the stretching of the string, and at one stage a certain term is assumed to be constant and named T. If you teach this course with a term-paper requirement, which I have done with some success, some students may take this as a starting point to investigate other derivations for the motion of elastic media.

Derivations are time-consuming (when done convincingly), so normally I only do one, either this one or the one for the heat equation. If you choose not to do this one, make sure that the students are aware that it is available.

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